Theorem carden2b 8391

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8390 are meant to replace carden 8965 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	|- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 8389	. . . . 5 |- ( ( card ` B ) e. ( card ` A ) -> -. ( card ` B ) ~~ A )
2		ennum 8371	. . . . . . . 8 |- ( A ~~ B -> ( A e. dom card <-> B e. dom card ) )
3	2	biimpa 486	. . . . . . 7 |- ( ( A ~~ B /\ A e. dom card ) -> B e. dom card )
4		cardid2 8377	. . . . . . 7 |- ( B e. dom card -> ( card ` B ) ~~ B )
5	3, 4	syl 17	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ B )
6		ensym 7616	. . . . . . 7 |- ( A ~~ B -> B ~~ A )
7	6	adantr 466	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> B ~~ A )
8		entr 7619	. . . . . 6 |- ( ( ( card ` B ) ~~ B /\ B ~~ A ) -> ( card ` B ) ~~ A )
9	5, 7, 8	syl2anc 665	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ A )
10	1, 9	nsyl3 122	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` B ) e. ( card ` A ) )
11		cardon 8368	. . . . 5 |- ( card ` A ) e. On
12		cardon 8368	. . . . 5 |- ( card ` B ) e. On
13		ontri1 5467	. . . . 5 |- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) ) )
14	11, 12, 13	mp2an 676	. . . 4 |- ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) )
15	10, 14	sylibr 215	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) C_ ( card ` B ) )
16		cardne 8389	. . . . 5 |- ( ( card ` A ) e. ( card ` B ) -> -. ( card ` A ) ~~ B )
17		cardid2 8377	. . . . . 6 |- ( A e. dom card -> ( card ` A ) ~~ A )
18		id 23	. . . . . 6 |- ( A ~~ B -> A ~~ B )
19		entr 7619	. . . . . 6 |- ( ( ( card ` A ) ~~ A /\ A ~~ B ) -> ( card ` A ) ~~ B )
20	17, 18, 19	syl2anr 480	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) ~~ B )
21	16, 20	nsyl3 122	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` A ) e. ( card ` B ) )
22		ontri1 5467	. . . . 5 |- ( ( ( card ` B ) e. On /\ ( card ` A ) e. On ) -> ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) ) )
23	12, 11, 22	mp2an 676	. . . 4 |- ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) )
24	21, 23	sylibr 215	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) C_ ( card ` A ) )
25	15, 24	eqssd 3478	. 2 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) = ( card ` B ) )
26		ndmfv 5896	. . . 4 |- ( -. A e. dom card -> ( card ` A ) = (/) )
27	26	adantl 467	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = (/) )
28	2	notbid 295	. . . . 5 |- ( A ~~ B -> ( -. A e. dom card <-> -. B e. dom card ) )
29	28	biimpa 486	. . . 4 |- ( ( A ~~ B /\ -. A e. dom card ) -> -. B e. dom card )
30		ndmfv 5896	. . . 4 |- ( -. B e. dom card -> ( card ` B ) = (/) )
31	29, 30	syl 17	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` B ) = (/) )
32	27, 31	eqtr4d 2464	. 2 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = ( card ` B ) )
33	25, 32	pm2.61dan 798	1 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1867   C_ wss 3433   (/)c0 3758   class class class wbr 4417   dom cdm 4845   Oncon0 5433   ` cfv 5592   ~~ cen 7565   cardccrd 8359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-er 7362  df-en 7569  df-card 8363

This theorem is referenced by:  card1  8392  carddom2  8401  cardennn  8407  cardsucinf  8408  pm54.43lem  8423  nnacda  8620  ficardun  8621  ackbij1lem5  8643  ackbij1lem8  8646  ackbij1lem9  8647  ackbij2lem2  8659  carden  8965  r1tskina  9196  cardfz  12169